Bounding the graviton mass using non-linear density wave theory

In this paper we use the Newtonian gravitational potential corrected by non-liner effects to obtain new bounds on graviton mass using non-linear density wave theory (NLDW). This potential differs from the gravitational potential obtained in other modified gravity theories (e.g. the weak field limit of Yukawa gravity, Modified Newtonian Dynamics, non-local theories, $\Lambda$ cold dark matter..). Using this model, we are able to define wavelength of the non-linear wave as an analytical solution of integrable non-linear differential equation (namely, non-linear Schrodinger equation). Assuming that the wavelength of the non-linear wave represents the graviton Compton wavelength, we have found the corresponding upper bound of graviton mass. We compare obtained result with first assessments of LIGO $\&$ Virgo collaboration and we find they are in a good agreement. Present model used to determine the upper limit of graviton mass is completely independent from other methods published until now. We have compared our result with results obtained using several chosen published methods.